Page 1
Optimal execution strategies
in limit order books
with general shape functions
Aur´ elien Alfonsi∗
CERMICS, projet MATHFI
Ecole Nationale des Ponts et Chauss´ ees
6-8 avenue Blaise Pascal
Cit´ e Descartes, Champs sur Marne
77455 Marne-la-vall´ ee, France
alfonsi@cermics.enpc.fr
Antje Fruth
Quantitative Products Laboratory
Alexanderstr. 5
10178 Berlin, Germany
fruth@math.tu-berlin.de
Alexander Schied∗
Department of Mathematics, MA 7-4
TU Berlin
Strasse des 17. Juni 136
10623 Berlin, Germany
schied@math.tu-berlin.de
To appear in Quantitative Finance
Submitted September 3, 2007, accepted July 24, 2008
This version: November 20, 2009
Abstract: We consider optimal execution strategies for block market orders placed in a limit
order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005)
but allow for a general shape of the LOB defined via a given density function. Thus, we can allow
for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We
distinguish two possibilities for modeling the resilience of the LOB after a large market order:
the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the
exponential recovery of the bid-ask spread. We consider both of these resilience modes and, in
each case, derive explicit optimal execution strategies in discrete time. Applying our results to
a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a
risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang
(2005). We also provide some evidence for the robustness of optimal strategies with respect to
the choice of the shape function and the resilience-type.
∗Supported by Deutsche Forschungsgemeinschaft through the Research Center Matheon “Mathematics for
key technologies” (FZT 86).
1
hal-00166969, version 3 - 3 Feb 2010
Page 2
1 Introduction.
A common problem for stock traders consists in unwinding large block orders of shares, which
can comprise up to twenty percent of the daily traded volume of shares. Orders of this size
create significant impact on the asset price and, to reduce the overall market impact, it is
necessary to split them into smaller orders that are subsequently placed throughout a certain
time interval. The question at hand is thus to allocate an optimal proportion of the entire order
to each individual placement such that the overall price impact is minimized.
Problems of this type were investigated by Bertsimas and Lo [8], Almgren and Chriss [3, 4],
Almgren and Lorenz [5], Obizhaeva and Wang [16], and Schied and Sch¨ oneborn [18, 19] to
mention only a few. For extensions to situations with several competing traders, see [11], [12],
[20], and the references therein.
The mathematical formulation of the corresponding optimization problem relies first of all
on specifying a stock price model that takes into account the often nonlinear feedback effects
resulting from the placement of large orders by a ‘large trader’. In the majority of models
in the literature, such orders affect the stock price in the following two ways. A first part of
the price impact is permanent and forever pushes the price in a certain direction (upward for
buy orders, downward for sell orders). The second part, which is usually called the temporary
impact, has no duration and only instantaneously affects the trade that has triggered it. It
is therefore equivalent to a (possibly nonlinear) penalization by transaction costs. Models of
this type underlie the above-mentioned papers [8], [3], [4], [5], [11], [12], and [20]. Also the
market impact models described in Bank and Baum [7], Cetin et al. [13], Frey [14], and Frey
and Patie [15] fall into that category. While most of these models start with the dynamics of
the asset price process as a given fundamental, Obizhaeva and Wang [16] recently proposed
a market impact model that derives its dynamics from an underlying model of a limit order
book (LOB). In this model, the ask part of the LOB consists of a uniform distribution of shares
offered at prices higher than the current best ask price. When the large trader is not active,
the mid price of the LOB fluctuates according to the actions of noise traders, and the bid-ask
spread remains constant. A buy market order of the large trader, however, consumes a block
of shares located immediately to the right of the best ask and thus increase the ask price by a
linear proportion of the size of the order. In addition, the LOB will recover from the impact of
the buy order, i.e., it will show a certain resilience. The resulting price impact will neither be
instantaneous nor entirely permanent but will decay on an exponential scale.
The model from [16] is quite close to descriptions of price impact on LOBs found in empirical
studies such as Biais et al. [9], Potters and Bouchaud [17], Bouchaud et al. [10], and Weber
and Rosenow [21]. In particular, the existence of a strong resilience effect, which stems from
the placement of new limit orders close to the bid-ask spread, seems to be a well established
fact, although its quantitative features seem to be the subject of an ongoing discussion.
In this paper, we will pick up the LOB-based market impact model from [16] and generalize
it by allowing for a nonuniform price distribution of shares within the LOB. The resulting
LOB shape which is nonconstant in the price conforms to empirical observations made in
[9, 17, 10, 21]. It also leads completely naturally to a nonlinear price impact of market orders
as found in an empirical study by Almgren et al. [6]; see also Almgren [2] and the references
therein. In this generalized model, we will also consider the following two distinct possibilities
for modeling the resilience of the LOB after a large market order: the exponential recovery
2
hal-00166969, version 3 - 3 Feb 2010
Page 3
of the number of limit orders, i.e., of the volume of the LOB (Model 1), or the exponential
recovery of the bid-ask spread (Model 2). While one can imagine also other possibilities, we
will focus on these two obvious resilience modes. Note that we assume the LOB shape to be
constant in time. Having a time-varying LOB shape will be an area of ongoing research.
We do not have a classical permanent price impact in our model for the following reasons:
Adding classical permanent impact, which is proportional to the volume traded, would be
somewhat artificial in our model. In addition, this would not change optimal strategies as the
optimization problem will be exactly the same as without permanent impact. What one would
want to have instead is a permanent impact with a sensible meaning in the LOB context. But
this would bring substantial difficulties in our derivation of optimal strategies.
After introducing the generalized LOB with its two resilience modes, we consider the prob-
lem of optimally executing a buy order for X0shares within a certain time frame [0,T]. The
focus on buy orders is for the simplicity of the presentation only, completely analogous results
hold for sell orders as well. While most other papers, including [16], focus on optimization
within the class of deterministic strategies, we will here allow for dynamic updating of trad-
ing strategies, that is, we optimize over the larger class of adapted strategies. We will also
allow for intermediate sell orders in our strategies. Our main results, Theorem 4.1 and Theo-
rem 5.1, will provide explicit solutions of this problem in Model 1 and Model 2, respectively.
Applying our results to a block-shaped LOB, we obtain a new closed-form representation for
the corresponding optimal strategy, which explicitly solves the recursive scheme given in [16].
Looking at several examples, we will also find some evidence for the robustness of the optimal
strategy. That is the optimal strategies are qualitatively and quantitatively rather insensitive
with respect to the choice of the LOB shape. In practice, this means that we can use them
even though the LOB is not perfectly calibrated and has a small evolution during the execution
strategy.
The model we are using here is time homogeneous: the resilience rate is constant and trading
times are equally spaced. By using the techniques introduced in our subsequent paper [1], it
is possible to relax these assumptions and to allow for time inhomogeneities and also for linear
constraints, at least in block-shaped models.
The method we use in our proofs is different from the approach used in [16]. Instead of
using dynamic programming techniques, we will first reduce the model of a full LOB with
nontrivial bid-ask spreads to a simplified model, for which the bid-ask spreads have collapsed
but the optimization problem is equivalent. The minimization of the simplified cost functional
is then reduced to the minimization of certain functions that are defined on an affine space.
This latter minimization is then carried out by means of the Lagrange multiplier method and
explicit calculations.
The paper is organized as follows. In Section 2, we explain the two market impact models
that we derive from the generalized LOB model with different resilience modes. In Section 3, we
set up the resulting optimization problem. The main results for Models 1 and 2 are presented
in the respective Sections 4 and 5. In Section 6, we consider the special case of a uniform
distribution of shares in the LOB as considered in [16]. In particular, we provide our new
explicit formula for the optimal strategy in a block-shaped LOB as obtained in [16]. Section 7
contains numerical and theoretical studies of the optimization problem for various nonconstant
shape functions. The proofs of our main results are given in the remaining Sections A through D.
More precisely, in Section A we reduce the optimization problem for our two-sided LOB models
3
hal-00166969, version 3 - 3 Feb 2010
Page 4
to the optimization over deterministic strategies within a simplified model with a collapsed
bid-ask spread. The derivations of the explicit forms of the optimal strategies in Models 1 and
2 are carried out in the respective Sections B and C. In Section D we prove the results for
block-shaped LOBs from Section 6.
2 Two market impact models with resilience.
In this section, we aim at modeling the dynamics of a LOB that is exposed to repeated market
orders by a large trader. The overall goal of the large trader will be to purchase a large amount
X0> 0 of shares within a certain time period [0,T]. Hence, emphasis is on buy orders, and we
concentrate first on the upper part of the LOB, which consists of shares offered at various ask
prices. The lowest ask price at which shares are offered is called the best ask price.
Suppose first that the large trader is not active, so that the dynamics of the limit order
book are determined by the actions of noise traders only. We assume that the corresponding
unaffected best ask price A0is a martingale on a given filtered probability space (Ω,(Ft),F,P)
and satisfies A0
This assumption includes in particular the case in which A0is a
Bachelier model, i.e., A0
We emphasize, however, that we can take any martingale and hence use, e.g., a geometric
Brownian motion, which avoids the counterintuitive negative prices of the Bachelier model.
Moreover, we can allow for jumps in the dynamics of A0so as to model the trading activities
of other large traders in the market. In our context of a risk-neutral investor minimizing the
expected liquidation cost, the optimal strategies will turn out to be deterministic, due to the
described martingale assumption.
Above the unaffected best ask price A0
available shares in the LOB: the number of shares offered at price A0
for a continuous density function f : R −→]0,∞[. We will say that f is the shape function of
the LOB. The choice of a constant shape function corresponds to the block-shaped LOB model
of Obizhaeva and Wang [16].
The shape function determines the impact of a market order placed by our large trader.
Suppose for instance that the large trader places a buy market order for x0> 0 shares at time
t = 0. This market order will consume all shares located at prices between A0and A0+ DA
where DA
?DA
Consequently, the ask price will be shifted up from A0to
0= A0.
t= A0+ σWtfor an (Ft)-Brownian motion W, as considered in [16].
t, we assume a continuous ask price distribution for
t+ x is given by f(x)dx
0+,
0+is determined by
0+
0
f(x)dx = x0.
A0+:= A0+ DA
0+;
see Figure 1 for an illustration.
Let us denote by Atthe actual ask price at time t, i.e., the ask price after taking the price
impact of previous buy orders of the large trader into account, and let us denote by
DA
t:= At− A0
t
4
hal-00166969, version 3 - 3 Feb 2010
Page 5
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
???? ?? ????????
?????? ?? ?????
? ???
???
?
??
?
?
?
??
?
??
?
?
?
??
?
??
?
?
Figure 1: The impact of a buy market order of x0shares .
the extra spread caused by the actions of the large trader. Another buy market order of xt> 0
shares will now consume all the shares offered at prices between Atand
At+:= At+ DA
t+− DA
t= A0
t+ DA
t+,
where DA
t+is determined by the condition
?DA
t+
DA
t
f(x)dx = xt. (1)
Thus, the process DAcaptures the impact of market orders on the current best ask price.
Clearly, the price impact DA
f is constant between DA
functions; see, e.g., Almgren [2] and Almgren et al. [6] for a discussion.
Another important quantity is the process
t+− DA
tand DA
t will be a nonlinear function of the order size xt unless
t+. Hence, our model includes the case of nonlinear impact
EA
t=
?DA
t
0
f(x)dx, (2)
of the number of shares ‘already eaten up’ at time t. It quantifies the impact of the large trader
on the volume of the LOB. By introducing the antiderivative
?z
of f, the relation (2) can also be expressed as
F(z) =
0
f(x)dx (3)
EA
t= F(DA
t) andDA
t= F−1(EA
t),(4)
where we have used our assumption that f is strictly positive to obtain the second identity.
The relation (1) is equivalent to
EA
t+= EA
t+ xt. (5)
5
hal-00166969, version 3 - 3 Feb 2010
Page 26
Proof: The structure of the proof is similar to the one of Theorem 4.1 although the computations
are different. Thanks to Lemma C.1, we know that there exists an optimal strategy ξ∗=
(x∗
0,...,x∗
N) ∈ Ξ. There also exists a corresponding Lagrange multiplier ν such that
∂
∂xiC(2)(x∗
From (56), we get
?F−1(x∗
Since h2 is one-to one, this implies in particular that x∗
0,...,N − 1. It follows from (23) also Dti+= F−1(x∗
Dti+= Dt0+= F−1(x∗
0,...,x∗
N) = ν,i = 0,...,N.
ν = h2
i+ F (Dti))?,i = 0,...,N − 1.
i+ F (Dti) does not depend on i =
i+ F (Dti)) is constant in i, and so
Dti+1= aF−1(x∗
0) and
0). (59)
Hence,
x∗
x∗
x∗
N
0
= F?h−1
= X∗
2(ν)?,
0− x∗
i
= x∗
0− F(Dti) = x∗
0− F?aF−1(x∗
0)?
for i = 1,...,N − 1, (60)
0− (N − 1)?x∗
N) is equal to
0− F?aF−1(x∗
0)??.
These equations link all market orders to the initial trade x∗
we find that C(2)(x∗
0) := C(2)?
=NG(x∗
0. Using (60) and once again (59),
0,...,x∗
C(2)
0(x∗
x∗
0,x∗
0) −?F?aF−1(x∗
0− F(aF−1(x∗
0)),...,X0− Nx∗
0)??
0+ (N − 1)F(aF−1(x∗
0))
?
?
+ G?X0+ N?F?aF−1(x∗
0
and thus
?
0)?− x∗
0(x∗
??
0
??.
The initial trade x∗
0must clearly be a local minimum of C(2)
?
which is equivalent to
∂
∂yC(2)
0) = 0. Therefore,
0 = ND0+− a2D0+f(Dt1)
f(D0+)+ DtN+
af(Dt1)
f(D0+)− 1,
DtN+= D0+f(D0+) − a2f(Dt1)
f(D0+) − af(Dt1).(61)
This is just equation (21), which has at most one solution, due to Lemma C.3. This concludes
the proof of the existence and the representation of the optimal strategy ξ(2)in Theorem 5.1.
Finally, we need to show the strict positivity of the optimal strategy. Thanks to the posi-
tivity of the optimal x∗
0, we get
x∗
i= x∗
0− F(aF−1(x∗
0)) > 0
for i = 1,...,N −1. So it only remains to show that x∗
DtN+= D0+f(D0+) − a2f(aD0+)
f(D0+) − af(aD0+)= D0+
The fraction on the right is strictly positive due to Lemma C.3. Hence,
DtN+> D0+=1
N> 0. We infer from (61) and (59) that
1 +af(aD0+) − a2f(aD0+)
f(D0+) − af(aD0+)
?
?
.
aDtN> DtN,
which implies x∗
N> 0.
26
hal-00166969, version 3 - 3 Feb 2010
Page 27
D Optimal strategy for block-shaped LOB.
Here we prove the results of Section 6.
Our aim is to prove Proposition 6.2, i.e., to show that the strategy (27) satisfies the re-
cursion (31). The key point is that we have indeed explicit formulas for the coefficients in the
backward schemes of Proposition 6.2.
Lemma D.1 The coefficients αn, βn, and γnfrom (33) are explicitly given by
αn =
(1 + a−1) − qλ[(N − n)(a−1− 1) + 2(1 + a−1)]
2q[(N − n)(a−1− 1) + (1 + a−1)]
1 + a−1
[(N − n)(a−1− 1) + (1 + a−1)]
(N − n)(1 − a−1)
2κ[(N − n)(a−1− 1) + (1 + a−1)].
(62)
βn =
γn =
The explicit form of the sequences δn, ǫnand φnfrom (32) is
δn =
2a−2[(N − n)(a−1− 1) + (1 + a−1)]
κ[(N − n)(1 − a−2) + (N − n + 2)(a−3− a−1)]
κ(a−1− a)
[(N − n)(a−1− 1) + (1 + a−1)]
(N − n + 1)(a−1− a) − (N − n)(1 − a2)
[(N − n)(a−1− 1) + (1 + a−1)]
(63)
ǫn =
φn =
.
This Lemma can be proved in two steps. First, by a backward induction, we get the explicit
formulas for α, β and γ. Then, combining (62) with (33) and (32), we get (63).
Proof of Proposition 6.2. We can deduce the following formulas from the preceding lemma:
δnǫn=
2
(N − n)(1 − a) + 2,δnφn=2
κ
(N − n)(1 − a) + 1
(N − n)(1 − a) + 2. (64)
They will turn out to be convenient in (31).
Let us now consider the optimal strategy (ξ∗
processes Dt:= DA
0,...,ξ∗
N) from (27). We consider the associated
tand Xtas defined in (28) and (30). For n = 0, we have
ξ∗
0=
X0
(N − 1)(1 − a) + 2=12δ1ǫ1
and it satisfies (31) because D0= 0. For n ≥ 1, we can show easily by induction on n that
Dtn= aκξ∗
0. From (27), we get that ξ∗
n= (1 − a)ξ∗
0for n ?∈ {0,N}, and therefore we get
0− (n − 1)(1 − a)ξ∗
Xtn= X0− ξ∗
0= [(N − n)(1 − a) + 1]ξ∗
0.
Using these formulas, and combining with (64), it is now easy to check that
for n ∈ {1,...,N − 1},
ξ∗
2[δn+1ǫn+1Xtn− δn+1φn+1Dtn],
which shows that the optimal strategy given in (27) solves (31).
n=1
27
hal-00166969, version 3 - 3 Feb 2010
Page 28
Acknowledgement. Support from the Deutsche Bank Quantitative Products Laboratory
is gratefully acknowledged. The authors thank the Quantitative Products Group of Deutsche
Bank, in particular Marcus Overhaus, Hans B¨ uhler, Andy Ferraris, Alexander Gerko, and Chrif
Youssfi for stimulating discussions and useful comments (the statements in this paper, however,
express the private opinion of the authors and do not necessarily reflect the views of Deutsche
Bank). Moreover, it is a pleasure to thank Anna Obizhaeva and Torsten Sch¨ oneborn for helpful
comments on earlier versions of this paper.
References
[1] Alfonsi, A., Fruth, A., Schied, A. Constrained portfolio liquidation in a limit order book
model. Banach Center Publ. 83, 9-25 (2008).
[2] Almgren, R. Optimal execution with nonlinear impact functions and trading-enhanced
risk, Applied Mathematical Finance 10 , 1-18 (2003).
[3] Almgren, R., Chriss, N. Value under liquidation. Risk, Dec. 1999.
[4] Almgren, R., Chriss, N. Optimal execution of portfolio transactions. J. Risk 3, 5-39
(2000).
[5] Almgren, R., Lorenz, J. Adaptive arrival price. In: Algorithmic Trading III: Precision,
Control, Execution, Brian R. Bruce, editor, Institutional Investor Journals (2007).
[6] Almgren, R., Thum, C. Hauptmann, E., Li, E. Equity market impact. Risk, July (2005).
[7] Bank, P., Baum, D. Hedging and portfolio optimization in financial markets with a large
trader. Math. Finance 14, no. 1, 1–18 (2004).
[8] Bertsimas, D., Lo, A. Optimal control of execution costs. Journal of Financial Markets,
1, 1-50 (1998).
[9] Biais, B., Hillion, P., Spatt, C. An empirical analysis of the limit order book and order
flow in Paris Bourse. Journal of Finance 50, 1655-1689 (1995).
[10] Bouchaud, J. P., Gefen, Y., Potters, M. , Wyart, M. Fluctuations and response in finan-
cial markets: the subtle nature of ‘random’ price changes. Quantitative Finance 4, 176
(2004).
[11] Brunnermeier, M., Pedersen, L. Predatory trading. Journal of Finance 60, 1825-1863
(2005).
[12] Carlin, B., Lobo, M., Viswanathan, S. Episodic liquidity crises: Cooperative and preda-
tory trading. Forthcoming in Journal of Finance.
[13] Cetin, U., Jarrow, R., Protter, P. Liquidity risk and arbitrage pricing theory. Finance
Stoch. 8 , no. 3, 311–341 (2004).
28
hal-00166969, version 3 - 3 Feb 2010
Page 29
[14] Frey, R. Derivative asset analysis in models with level-dependent and stochastic volatility.
Mathematics of finance, Part II. CWI Quarterly 10 , no. 1, 1–34 (1997).
[15] Frey, R., Patie, P. Risk management for derivatives in illiquid markets: a simulation
study. Advances in finance and stochastics, 137–159, Springer, Berlin, 2002.
[16] Obizhaeva, A., Wang, J. Optimal Trading Strategy and Supply/Demand Dynamics,
Preprint, forthcoming in Journal of Financial Markets.
[17] Potters, M., Bouchaud, J.-P. More statistical properties of order books and price impact.
Physica A 324, No. 1-2, 133-140 (2003).
[18] Schied, A., Sch¨ oneborn, T. Optimal basket liquidation with finite time horizon for CARA
investors. Preprint, TU Berlin (2008).
[19] Schied, A., Sch¨ oneborn, T. Risk aversion and the dynamics of optimal liquidation strate-
gies in illiquid markets. To appear in Finance and Stochastics.
[20] Sch¨ oneborn, T., Schied, A. Competing players in illiquid markets: predatory trading vs.
liquidity provision. Preprint, TU Berlin.
[21] Weber, P., Rosenow, B. Order book approach to price impact. Quantitative Finance 5,
no. 4, 357-364 (2005).
29
hal-00166969, version 3 - 3 Feb 2010